Loehle and Scafetta Thinking Scientist
Hi Craig, always interested to see your work. I have a slightly different take on how I interpret the findings. You are trying to decompose a time series into statistically plausible parts and by doing so providing bounds on the interpretation of the magnitude of those parts. I do not need to believe whether the additional linear trend after 1942 is AGW or something else, by fitting the model in the way you have you can attempt to place a bound on the possible increase per century if that trend were interpreted in that way.
The periodic functions you fit are fairly clear – I have run my own tests using GISS (up to 2008) to look for the sine wave function with the best fit. To the nearest decade it is 60 years (I actually get a best fit of 66 years to GISS), so my simple test agrees with yours. I did not look for a further component by subtracting the 60 year cycle and fitting a periodic function to the residuals.
Instead what I did was take your result from Loehle 2007 which gives a 2000 year reconstruction and perform the same analysis – what is the sine wave with the best fit to the 2,000 year data? The answer is about 1560 years (which is where Singers’ 1,500 year cycle comes from).
If you plot the 1560 years sine wave which best correlates with Loehle 2007 you get an upswing in the GISS data period. Because of the long period of this 1560 year sine wave component, the part of the sine wave over the data length of GISS is indistinguishable from a straight line (OLS linear regression of the sine wave over the GISS time period gives R^2 = 0.9999). This is as valid an explanation of the apparent upward trend in the modern era without invoking any other mechanism to explain a linear trend. Now postulate just one other long period trend (200 years? 400 years?) and you can have a high degree of fit to the modern temperature record with a long term process represented by a periodic but stationary sine wave. Any sine wave with a period longer than the length of the time series (in the case of GISS, about 130 years) will give a fit almost indistinguishable from a straight line fit. Using just 4 sine waves in a periodic but stationary function will give a pretty good fit to the modern day temperature decomposition you showed without invoking any AGW at all.